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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 陳其誠 | |
| dc.contributor.author | Zong-Yi Pan | en |
| dc.contributor.author | 潘宗驛 | zh_TW |
| dc.date.accessioned | 2021-06-16T23:07:36Z | - |
| dc.date.available | 2012-08-10 | |
| dc.date.copyright | 2012-08-10 | |
| dc.date.issued | 2012 | |
| dc.date.submitted | 2012-08-04 | |
| dc.identifier.citation | [1] J-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no.4, 429-440.
[2] Paulo Ribenboim, Classical Theory of Algebraic Numbers, Springer, New York, 2001 [3] Marcus, Daniel A. Number Fields. Universitext. Springer-Verlag, New York-Heidelberg, 1977. [4] I. M. Isaacs, Finite Group Theory, American Mathematical Society v.92, 2008. [5] D. S. Dummit and R. M. Foote, Abstract Algebra. John Wile & Sons, Inc., New York, third edition, 2004. [6] F. Diamond, J.Shurman, A rst course in modular forms. Springer, 2005. [7] Miyake Toshitsune. Modular Forms, Translated from the Japanese by Yoshitaga Maeda. Springer-Verlag, Berlin, 1989 [8] J. Neukirch, Algebraic Number Theory, Translated from the 1992 German original and with a note by Norbert Schappacher. With a foreword by G. Harder. Grundlehren der Mathematischen Wissenschaften, 322. Springer-Verlag, Berlin, 1999. [9] Brauer, Richard, On Artin's L-series with general group character, Ann. of Math. (2) 48, 1947. [10] H. W. Lenstra, Jr., and P. Stevenhagen, Chebotar ev and his density theorem, Math. Intelligencer 18 (1996), 26-37. MR 97e:11144 [11] Charles W. Curtis, I. Reiner, Methods of Representation Theory with Applications to Finite Groups and Orders', Wiley, New York, 1981 [12] Cox, David A. Primes of the form x2 + ny2, Fermat, class eld theory and complex multiplication. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1989 [13] D. S. Dummit, Solving solvable quintics, Math. Comp. 57(1991), 387-401 [14] N. Koblitz, p-adic numbers, p-adic analysis, and zeta functions. Springer, Berlin, 1984. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/64915 | - |
| dc.description.abstract | J.P. Serre 在 [1] 證明了有關不可約多項式模掉質數 $p$ 之解個數的定理。本篇論文開頭描述此定理並且給完整的證明。論文第二部分將以一個五次多項式 x^5-5x+12 為例來探討並且找出此多項式對應的 Artin L-函數與 weight 為 1 、 level 為 1000的模型式之間的關係。 | zh_TW |
| dc.description.abstract | In [1], J.P. Serre proves a theorem concerning the numbers of solutions to irreducible
polynomial modulo prime numbers. In this thesis, we rst study the theorem and give a detailed proof, and in the second part of this thesis, we study the example of the polynomial x55x+12 and relate the Artin L-function of the splitting eld to modular forms in M1(0(1000)). | en |
| dc.description.provenance | Made available in DSpace on 2021-06-16T23:07:36Z (GMT). No. of bitstreams: 1 ntu-101-R99221017-1.pdf: 825977 bytes, checksum: 17b35c6fd72785a863d1c0b6e0a78f1d (MD5) Previous issue date: 2012 | en |
| dc.description.tableofcontents | Contents
Acknowledgements i Abstract (in Chinese) ii Abstract (in English) iii Contents iv List of Tables v 1 Introduction 1 2 Notation & Basic knowledge 2 2.1 Primary Decomposition in Galois extension . . . . . . . . . . . . . . . . . . . . . 2 2.2 Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Representations of Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Artin L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Adeles Rings & Ideles Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.6 Hecke L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.7 Modular Forms and Hecke Operator . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 The Proof of the Main Theorem 19 3.1 Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 The Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 An Example 22 4.1 The Galois Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 The Expression of Np(f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.3 The Corresponding Hecke Characters . . . . . . . . . . . . . . . . . . . . . . . . 23 4.4 The Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.5 The Frobenius Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 References 32 | |
| dc.language.iso | en | |
| dc.subject | 伽羅瓦表現 | zh_TW |
| dc.subject | 模型式 | zh_TW |
| dc.subject | Modular Forms | en |
| dc.subject | On A Theorem of Jordan | en |
| dc.subject | Galois Representations | en |
| dc.title | 不可約多項式模p解之探討 | zh_TW |
| dc.title | A Survey on the Solutions to Irreducible Polynomials Modulo p and Modular Forms | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 100-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 紀文鎮,黃柏嶧 | |
| dc.subject.keyword | 伽羅瓦表現,模型式, | zh_TW |
| dc.subject.keyword | On A Theorem of Jordan,Galois Representations,Modular Forms, | en |
| dc.relation.page | 32 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2012-08-06 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 數學研究所 | zh_TW |
| 顯示於系所單位： | 數學系 | |
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